The specific heat of tin and gallium in their stable and undercooled pure liquid states

THE

SPECIFIC

HEAT

OF

TIN

AND

GALLIUM

UNDERCOOLED

PURE

H.

and

S. CHENt

LIQUID

IN

THEIR

STABLE

AND

STATES*

D. TURNBULLt

range from The specific heats, C,, of tin and gallium in their pure liquid state over the temperature 632 for tin and 611 for gallium to 425 and 245”K, corresponding to undercoolings of 80 and 58°K Over the entire temperature range, the C, for both tin and gallium falls respectively have been measured. continuously with increasing temperature through their melting point with no singular behavior there. The excess specific heat, AC,, of the liquid over the corresponding crystal diverges with falling temperature from -0.38 for tin and 0.56 for gallium at their melting point to +0.27 and 1.04 Cal/g-atm-“K at the lowest temperature to which the particular specimen could be undercooled. For both tin and gallium, the specific heat at constant volume, C,, of the liquids at their melting point exceeds the Dulong The excess specific heat, AC,, as well as Petit limit value of 3R, and rises with decreasing temperature. AC”,, of liquid over the corresponding crystal in the undercooled region is attributed to the taking up of configurational entropy with rising temperature. CHALEUR

SPECIFIQUE

DE

L’ETAIN STABLE

ET ET

DU GALLIUM SURFONDU

A

L’ETAT

LIQUIDE

PUR

On mesure les chaleurs specifiques C, de l’etain et du gallium a l’etat liquide pur dans le domaine de temperatures allant de 632 pour l’etain et 611 pour le gallium a 425 et 245 “K, ce qui correspond a des surfusions de 80 et 58°K respectivement. Dans tout le domaine de temperatures, le C, pour Q la fois l’etain et le gallium diminue continuellement lorsque la temperature croit en traversant le point de fusion et il n’y a pas de singularite en ce point. L’exces de chaleur specifique, AC,, du liquide sur le cristal correspondant varie, quand on abaisse la temperature, des -0,38 pour l’etain et 0,56 pour le gallium, au point de fusion a +0,27 et 1,04 Cal/g atm “K, a la temperature la plus basse a laquelle l’echantilPour l’etain et le gallium, la chaleur specifique a volume constant, G,, lon consider& peut 6tre surfondu. du liquide au point de fusion, depasse la limite 3R de Dulong-Petit, et croit lorsque la temperature decroit. L’exces de chaleur specifique, AC,, aussi bien que AC,, du liquide sur le cristal correspondent dans la region surfondue est attribue a l’apparition d’une entropie de configuration lorsque la temperature s‘eleve. DIE

SPEZIFISCHE WARME UNTERKUHLTES

VON ZINN UND GALLIUM IM REIN FLUSSIGEN ZUSTAND

STAB&EN

Die spezifische Warme C, von Zinn und Gallium im rein fliissigen Zustand wurde im Temperaturbereich 632 (Zinn) und 611 (Gallium) bis 425 und 245”K, was Unterkiihlungen von 80 bzw. 58°K entspricht, gemessen. Im gesamten Temperaturbereich nimmt C, von Zinn Gallium kontinuierlich mit zunehmender Temperatur iiber Schmelzpunkt hinweg, ohne Singularitat ab. Der Ifberschuss der spezifischen Warme AC, der Fliissigkeit gegeniiber dem entsprechenden Kristall variiert mit fallender Temperatur von -0,38 fur Zinn und 0,56 fur Gallium am Schmelzpunkt bis +0,27 und 1,04 Cal/g-atom-OK bei der tiefsten Temperatur, die bei der Unterkiihlung erreichbar war. Sowohl fur Zinn als such fur Gallium liegt die spezifische Warme bei konstantem Volumen C, der Fhissigkeit am Schmelpunkt oberhalb des Dulong Petit’schen Grenzwertes von 3R und nimmt mit abnehmender Temperatur zu. Der Uberschuss AC, und such AC, der Fliissigkeit gegeniiber dem Kristall im Unterktihlungsbereich wird der Zunahme der Konfigurationsentropie mit zunehmender Temperatur sugeschrieben.

INTRODUCTION

It is well established pure liquid metals Zn) falls over point.

temperature,

that the specific heat, C,, of

(Hg, Li, Na, K, In, Bi, Pb and

continuously

with

a wide temperature

solid form.

increasing

temperature

detail some measurements

range above

the melting

in the undercooled preliminarily

alloys in liquid

This paper describes

in

of C, of gallium and tin,

liquid state, which were reported

some time ago.f3)

region this rising trend of C, with falling EXPERIMENTAL

temperature extends and whether or not C, exhibits any singular behavior at the thermodynamic melting

The heat capacity was measured with a differential scanning calorimeter, Perkin-Elmer DSC- 1. This instrument measures the difference in electric power

* Received July 21, 1967. This research was supported in part by the Office of Naval Research (Contract Nonr 1966(50)) and by the Advanced Research Projects Agency (Contract ARPA SD-88). t Division of Engineering and Applied Physics, Harvard University, Cambridge, Massachusetts. ACTA

have already reportedos2) some

of C, on undercooled

and amorphous

It is of interest to ascertain how far into the

undercooled

T,. We

measurements

METALLURGICA,

VOL.

16, MARCH

1968

required to keep two well-matched

holders, one (the

reference holder) empty and the other containing sample, 369

at

the

same

temperature.

The

the

power

ACTA

370

difference, as the

Y, is recorded as a function

holders

(scanning

are

the reference

capacity

and thermal

consists

given instrument

rate,

holders,

of the difference

setting and scanning rate, the power

differentials.

differential

at a

Thus

thermal

and empty

to the isothermal

loss

Y,

(p = 0)

is a measure

rates

and

heat

of the

capacities

the two holders and this must be reproduced

from run to run in order for the Y, vs. T base line to be reproduced. covering

This

can be accomplished

both holders with snug-fitting

by:

(1)

covers, taking

ments

environment

The reproducibility

we reproduced at a fixed instrument setting.

The temperature

scale was calibrated

as standards

the

melting

within

points

51°C

of the

pure

capacity

weight.

(4 x 10-s cm3

and in our experi-

1.5 x 1O-3 to 3.0 x 1O-3

For the measurements were tantalum Aluminum

covers

were used

in measurements

droplet

samples

contact.

Before

was evacuated This

through

prevents

The samples 1.

the system oxidation

of the

(flow rate 10 ems/

of the samples

samples were freshly

metal.

The

from coalescing the droplets

droplets

and

convection.

tin droplet

samples

were

in air at 150°C for one hour.

were formed

this the droplets

are listed in cut from large

by a thin oxide film formed

gallium in a warm alcoholic After

thermal

the holder system

used in the experiments

“Bulk”

Gallium

and

on the

in order to insure optimum each measurement

provides good thermal Table

containers

and then dried high purity nitrogen gas

was circulated min).

(sealed)

on bulk

pans and the

covers were also tantalum.

by heating

of the specific heat

is assured only if the isothermal recorder

measurements

using

thermodynamic

around both holders at the corresponding

t)emperature. positions

the

from

the containers

prevented

reproduce

ranged

gram atomic

pieces

to

by the

of the sample holders

they

(2)

care

196X

are limited

care to position them in the same way from run to run, taking

16,

in volume)

samples

in first determining.

relative

VOL.

weights 9

loss rate between

Y,, vs. T between the reference

difference,

between

of temperature

a specified

at

and sample holders.

The procedure

power

heated

It (Y) is a measure

rate).

in the thermal

sample

METALLURGICA,

by dispersing

liquid

solution of sodium oleate.

were solidified and separated

from the solution. Four series of C, measurements made:

tial

excess specific heat, AC,, of liquid tin over that of the

between

containing

the

reference

the sample

specified

setting

as the sample.

recorder

positions

temperature

for

sapphire

therefore: where tively,

knownt4)

equilibrium

the

with

and blank. Y Y’ -

molar

precautions

heat

heat

in the measurements;

of the

depending C,’

of sapphire

of the

above

of &2%.

sample

temperature,

capacity C,

without

sample. may

a scanning

finest powders

and Chemical

(with particles

purity

bought

from

Co., N.Y. ; high purity

125 and 325 mesh powders obtained Chemical,

from

Liquid

tin

the finest powder would sustain,

appreciable (to 460°K)

purity

N.Y.

solidification,

undercoolings

for long periods.

droplets

would

of

However,

sustain

much

the

larger

and of

undercoolings

on the

of the 125 mesh powder and of the 325 mesh powders

is

the

solidified,

for long periods.

respectively,

420°K

With

was measured from 632 to 425°K.

the

by results

rate of lO”C/min

over temperature intervals of 10 to 15°C was used. In a few measurements on smaller samples the scanning rate was 20”C/min over 30°C intervals. The sample

(AT = 85“).

undercooling

of 80°K.

small fraction Gallium

Only small fractions

above 460°K

and C, is

be determined

This is attested

higher

the

35°C below the melting

10 p) of chemical

by melting

only 45°K

obtained (see results) on crystalline tin and gallium which are in good agreement with earlier values in the literature.(5,6) In most measurements

Drug

(5’9 purity)

Y&N

on

There were three sources of tin powders used

sizes less than

formed

and two in

measurements

to a temperature

at a given test

more

Liquid tin in bulk form could usually be

undercooled point.

the

in bulk form

were made with droplet samples in the under-

Gallard-Schlesinger

Y’ -

constant

Three

Y, be the

rate, N and N’ are, respec-

and

capacity

outlined

with an accuracy

using

Y, = kpC,‘N’

k is a calibration setting

crystal

cooled region.

A

Amend

Y,)N’C,‘/(

form.

rate.

Then,

of moles

droplet

T at the

Y, = k!!‘C,N

C, = (Y -

molar

Y’ and

respectively,

the number

instrument

Y,

holder

purposes,

at dynamic

i7 is the scanning

sapphire,

scanning

runs

for the sample: for sapphire:

Let

the

against

and

is made for calibration

sapphire

sample,

and

is measured

instrument

determination

holder

two with samples

on liquid tin were

metals, Hg, Ga; In, Sn, Pb and Zn. After establishing the base lines the power differen-

The

specific

Corrections

(AT = 45”) and heat

of liquid

corresponding

tin

to an

were made for the

of tin which had solidified.

melts in bulk form could occasionally

undercooled without solidification fraction of the liquid gallium

to 250°K. droplets

be

A small solidified

occasionally at 250”K, but the greatest portion of the droplets remained liquid over the entire period of observations the droplets

at temperatures above 240’K. Some of solidified to form metastable Ga-II phase

CHES

AND

TURSBULL:

SPECIFIC

HEAT

TABLE

OF

UNDERCOOLED

Sn

AND

371

Ga

1. Samples Weight

Element

Form

Purity

Source

(mg)

Sn-13

sn

bulk

5’9

Cominco American

199.00

C,

X11-22

Sn

bulk

5’9

Cominco American

209.04

ClJ

Sn-20

Sn

powder 325 mesh

5’9

GallardSchleinger

156.00

c,

Sn-21

Sn

powder 125 mesh

5’9

GallardSchleinger

136.00

(‘P

Sn-19

Sn

powder 325 mesh

5’9

GallardSchleinger

156.00

Su- I6

Sn

powder 125 mesh

5’9

GallardSchleinger

115.00

AC,

Sn 18

Sn

powder 325 mesh

5’9

GallardSchleinger

135.30

AC,

Ga-8

Ga

bulk

7’9

EaglePitcher

208.80

c,

Ga-13

Ga

bulk

7’9

EaglePi&her

208.80

c,

(~a-15

Ga

bulk

7’9

EaglePitcher

208.20

(‘0

Ga- 16

G&

droplet 200 p dia.

7’9

EaglePitcher

167.10

c9

which melts at 257”K.t’) corresponding

to an undercooling

specific

heat

of

the

states also has been measured 208°K

for gallium

AC,(C,z

-- C,“)

The specific heat of gallium RESULTS

in the liquid state was measured from 611 to 245”K, The

Measurement

of 58°K.

equilibrium

The specific heat, C, of the various phases, and the crystalline

from 327 for tin and

up to their melting

points,

re-

spectively.

excess specific heat, AC,,

of liquid over that of the

crystal for tin are plotted against temperature Over the entire temperature

range measured,

A

CP

zoo

#13:

0.0

#22:

A.d

G 20:

.I

+

031 gotm°K

21.

v

xv+

-

#19:. #16:x

300

400

T’K

500

in Fig.

1 and Fig. 2. Figure 3 shows the results for gallium.

600

FIG. 1. The specific heat, C,, of pure tin near the melting point, T, ( = 505°K). I designates the liquid, and z the crystalline phase. AC, = C,& - CDT. Only the difference in heat capacity between the liquid and crystalline states was measured on samples 16, 18 and 19 (see inset); the branch of the C,” curve beyond its intersection with the C,” curve is obtained from the smoothed curve through these data.

C,

372

ACTA

2.0

I.9

i I

VOL.

16,

-.-

PRESENI

.-.-

SELECTEG

1968

WOR4: VALIJE

(Whale)

,’

i ’

’ 0

METALLURGICA,

I

I

i FIG.

2.

volume,

r, I

I

I

100

200

xx?

I TOti

400

L

6

ii00

600

Summarized specific heat, C,, and calculated values of specific heat at constant C,, of pure tin. I designates the liquid and z the crystalline phase. Selected values are of Hultgren et aLf5

60-

B”IG. 3. The specific heat, C,, and calculated values of specific heat at constant volume, t design&es the liquid and z the crystalline phase (&-I). C*, of pure gallium. T,, (= 303’K) is t,he melting point of pure gallium.

CHEN

TURNBULL:

AND

SPECIFIC

HEAT

OF UNDERCOOLED

of the liquid continues to rise with decreasing tempera-

diverging

ture from 6.80 & 0.10 for tin and 6.40 f

Furthermore

atm-“K

for gallium

0.10 Cal/g-

to 7.26 + 0.10 and 7.05 + 0.10

Cal/g-atm-“K, respectively. The C, of the liquids are expressed empirically as follows: C, = 9.97 -

C, = 8.28 -

9.15 x 10-3T

6.10 x 10-3T f

The

specific

evaluated

heats

for

both

tin

for tin

and

for gallium

volume,

gallium

of the

C, were

following

the

According

effect,

Petit limit of 3R.

V is the molar volume.

the data of Gordon,(*)

book,

American (1963)

McGraw-Hill,

Institute and

N.Y.

of Physics,

International

which

is opposite

observed

(1928) were used.

McGraw-Hill, Tables,

The calculated

C, are shown in Fig. 2 and Fig. 3. The specific heats, C, of the crystalline states are in good agreement with the selected valuef5) for tin and that of Adams et aZ.t6) for gallium. INTERPRETATION

We noted that the specific heats, C, as well as C,, for both tin and gallium, continue

to rise through

singular behavior ture

to

which

undercooled.

in their pure liquid the melting

state,

point, with no

there, down to the lowest temperathe

particular

specimen

could

be

For both tin and gallium, the C,, as well

as C,, of the liquid

at the lowest

undercooling

the

calculations

and electronic to

for liquid

neglectDulong indicate

contributions

to

the

decreasing

trend

of

C,

metals with rising temperature.

We therefore attribute

the decreasing metals,

or AC,, of liquid

For

Hoather,tg)

Critical

reach

trend of C,, as

and the excess specific over the corresponding

region to the picking up of

entropy with rising temperature.

K

Hunter et cd.,(lO) and selected values from the HandN.Y.,

(e.g. Na,

the specific heat increase with increasing temperature,

configurational coefficient,

and

Theoretical

crystal in the undercooled

these calculations

crystal.

to the Debye model of specific heat, the

ing the anharmonic

heat, AC,

is the compressibility,

corresponding

C, of the liquids for both

Li, etc.) exceeds the Dulong Petit limit of 3R.

well as C,, of liquid

equation:

where B is the bulk thermal expansion

373

tin and gallium as well as for some metals

that the anharmonic

+ 5.0 x 10-6T2

at constant

that

the calculated

Ga

C, will increase with increasing temperature,

+ 6.5 x lo-V2 & 0.10 Cal/g-atm”K

0.10 Cal/g-atm”K

from

Sn AXD

is

ACKNOWLEDGMENT

The

authors

Bienenstock

are

indebted

to

Professors

A.

I.

and P. C. Martin for helpful discussions

on the theory for the heat capacity

of liquid metals.

REFERENCES 1. H. S. CHEN and D. TURNBULL, Appl. Phys. Lett. 10, 284 (1967). 2. H. S. CHEN and D. TURNBUI~L,J. qpl. Phys. 38, 3646 (1967). 3. H. S. CHEN and D. T~RNBULL, Bull. phys. Sot. 11, 326 (1966), Spring Meeting at Washington, D.C. 4. D. C. GINNINGS and G. T. FXYRUKAWA, J. Am. them. Sot. 75, 5.22 (1953). 5. R. HULTGIREN,R. L. ORR, P. D. ANDERSON, and K. K. KELLEY, #elected Values of Thermodynamic Properties of Metals and Alloys. Wiley (1963). 6. G. C. ADAMS, JR., H. L. JOHNSTONand E. C. KERR, J. Am. them. Sot. 74, 3784 (1952). 7. A. DEFRAIN, I. EPELBOIN, and M. ERUG, C.h. hebd. Sdanc. Acad. Sci., Paris 248, 1486 (1959). 8. R. B. GORDON, Aeta Met. 7, 1 (1959). 9. W. H. HOATRER, Proc. phys. Sot. 48, 699 (1936). 10. J. I,. HUNTER and K. S. HOVAN,J. them. Phys. 14,4013.